Integral Closure of Ideals, Rings, and Modules

454 Pages·2015·3.278 MB·English

by  Irena Swanson, Craig Huneke

London Mathematical Society
Lecture Note Series 336
Integral Closure
of
Ideals, Rings, and Modules
Irena Swanson and Craig Huneke
Cambridge University Press

London Mathematical Society Lecture Note Series 336
Integral Closure of Ideals,
Rings, and Modules
Craig Huneke
University of Kansas
Irena Swanson
Reed College, Portland
Cambridge University Press

Contents
Contents v
Table of basic properties ix
Notation and basic definitions xi
Preface xiii
1. What is integral closure of ideals? 1
1.1. Basic properties 2
1.2. Integral closure via reductions 5
1.3. Integral closure of an ideal is an ideal 6
1.4. Monomial ideals 9
1.5. Integral closure of rings 13
1.6. How integral closure arises 14
1.7. Dedekind–Mertens formula 17
1.8. Exercises 20
2. Integral closure of rings 23
2.1. Basic facts 23
2.2. Lying-Over, Incomparability, Going-Up, Going-Down 30
2.3. Integral closure and grading 33
2.4. Rings of homomorphisms of ideals 39
2.5. Exercises 42
3. Separability 48
3.1. Algebraic separability 48
3.2. General separability 49
3.3. Relative algebraic closure 53
3.4. Exercises 55
4. Noetherian rings 57
4.1. Principal ideals 57
4.2. Normalization theorems 58
4.3. Complete rings 61
4.4. Jacobian ideals 64
4.5. Serre’s conditions 71
4.6. Affine and Z-algebras 74
4.7. Absolute integral closure 78
4.8. Finite Lying-Over and height 80
4.9. Dimension one 84
4.10. Krull domains 87
4.11. Exercises 91
5. Rees algebras 96
5.1. Rees algebra constructions 96
5.2. Integral closure of Rees algebras 98
5.3. Integral closure of powers of an ideal 100
vi
5.4. Powers and formal equidimensionality 103
5.5. Defining equations of Rees algebras 107
5.6. Blowing up 111
5.7. Exercises 112
6. Valuations 116
6.1. Valuations 116
6.2. Value groups and valuation rings 118
6.3. More properties of valuation rings 120
6.4. Existence of valuation rings 123
6.5. Valuation rings and completion 124
6.6. Some invariants 127
6.7. Examples of valuations 132
6.8. Valuations and the integral closure of ideals 136
6.9. The asymptotic Samuel function 141
6.10. Exercises 142
7. Derivations 146
7.1. Analytic approach 146
7.2. Derivations and differentials 150
7.3. Exercises 152
8. Reductions 153
8.1. Basic properties and examples 153
8.2. Connections with Rees algebras 157
8.3. Minimal reductions 158
8.4. Reducing to infinite residue fields 162
8.5. Superficial elements 163
8.6. Superficial sequences and reductions 168
8.7. Non-local rings 172
8.8. Sally’s theorem on extensions 174
8.9. Exercises 176
9. Analytically unramified rings 180
9.1. Rees’s characterization 181
9.2. Module-finite integral closures 183
9.3. Divisorial valuations 185
9.4. Exercises 188
10. Rees valuations 190
10.1. Uniqueness of Rees valuations 190
10.2. A construction of Rees valuations 194
10.3. Examples 199
10.4. Properties of Rees valuations 204
10.5. Rational powers of ideals 208
10.6. Exercises 211
11. Multiplicity and integral closure 216
11.1. Hilbert–Samuel polynomials 216
11.2. Multiplicity 221
vii
11.3. Rees’s theorem 226
11.4. Equimultiple families of ideals 229
11.5. Exercises 236
12. The conductor 238
12.1. A classical formula 239
12.2. One-dimensional rings 239
12.3. The Lipman–Sathaye theorem 241
12.4. Exercises 246
13. The Brian¸con–Skoda Theorem 248
13.1. Tight closure 249
13.2. Brianc¸on–Skoda via tight closure 252
13.3. The Lipman–Sathaye version 254
13.4. General version 257
13.5. Exercises 260
14. Two-dimensional regular local rings 262
14.1. Full ideals 263
14.2. Quadratic transformations 268
14.3. The transform of an ideal 271
14.4. Zariski’s theorems 273
14.5. A formula of Hoskin and Deligne 279
14.6. Simple integrally closed ideals 282
14.7. Exercises 285
15. Computing integral closure 287
15.1. Method of Stolzenberg 288
15.2. Some computations 292
15.3. General algorithms 298
15.4. Monomial ideals 301
15.5. Exercises 303
16. Integral dependence of modules 309
16.1. Definitions 309
16.2. Using symmetric algebras 311
16.3. Using exterior algebras 314
16.4. Properties of integral closure of modules 316
16.5. Buchsbaum–Rim multiplicity 320
16.6. Height sensitivity of Koszul complexes 326
16.7. Absolute integral closures 328
16.8. Complexes acyclic up to integral closure 332
16.9. Exercises 334
17. Joint reductions 338
17.1. Definition of joint reductions 338
17.2. Superficial elements 340
17.3. Existence of joint reductions 342
17.4. Mixed multiplicities 345
17.5. More manipulations of mixed multiplicities 351
viii
17.6. Converse of Rees’s multiplicity theorem 355
17.7. Minkowski inequality 357
17.8. The Rees–Sally formulation and the core 360
17.9. Exercises 365
18. Adjoints of ideals 367
18.1. Basic facts about adjoints 367
18.2. Adjoints and the Brianc¸on–Skoda Theorem 369
18.3. Background for computation of adjoints 371
18.4. Adjoints of monomial ideals 373
18.5. Adjoints in two-dimensional regular rings 376
18.6. Mapping cones 379
18.7. Analogs of adjoint ideals 382
18.8. Exercises 383
19. Normal homomorphisms 385
19.1. Normal homomorphisms 386
19.2. Locally analytically unramified rings 388
19.3. Inductive limits of normal rings 390
19.4. Base change and normal rings 391
19.5. Integral closure and normal maps 395
19.6. Exercises 397
Appendix A. Some background material 399
A.1. Some forms of Prime Avoidance 399
A.2. Carath´eodory’s theorem 399
A.3. Grading 400
A.4. Complexes 401
A.5. Macaulay representation of numbers 402
Appendix B. Height and dimension formulas 404
B.1. Going-Down, Lying-Over, flatness 404
B.2. Dimension and height inequalities 405
B.3. Dimension formula 406
B.4. Formal equidimensionality 408
B.5. Dimension Formula 410
References 412
Index 430
Table of basic properties
In this table, R is an arbitrary Noetherian ring, and I,J ideals in R. The
overlines denote the integral closure in the ambient ring.
(1) I I √I and √0 I. (Page 2.)
⊆ ⊆ ⊆
(2) I = I. (Corollary 1.3.1.)
(3) Whenever I J, then I J. (Page 2.)
⊆ ⊆
(4) For any I,J, I : J I : J. (Page 7.)
⊆
(5) For any finitely generated non-zero I and any J in a domain,
IJ : I = J. (Corollary 6.8.7.)
(6) An intersectionof integrally closed ideals is an integrallyclosed
ideal. (Corollary 1.3.1.)
ϕ
(7) Persistence: if R S is a ring homomorphism, then ϕ(I)
−→ ⊆
ϕ(I)S. (Page 2.)
(8) If W is a multiplicatively closed set in R, then IW 1R =
−
IW 1R. (Proposition 1.1.4.)
−
(9) An element r R is in the integral closure of I if and only if
∈
for every minimal prime ideal P in R, the image of r in R/P
is in the integral closure of (I +P)/P. (Proposition 1.1.5.)
(10) Reduction criterion: J I if and only if there exists l
⊆ ∈
N such that (I +J)lI = (I +J)l+1. (Corollary 1.2.5.)
>0
(11) Valuative criterion: J I if and only if for every (Noe-
⊆
therian) valuation domain V which is an R-algebra, JV IV.
⊆
When R is an integral domain, the V need only vary over val-
uation domains in the field of fractions of R. Furthermore, the
V need only vary over valuation domains centered on maximal
ideals. (Theorem 6.8.3 and Proposition 6.8.4.)
(12) I J IJ. (Remark 1.3.2 (4), or Corollary 6.8.6.)
· ⊆
(13) Let R S be an integral extension of rings. Then IS R = I.
⊆ ∩
(Proposition 1.6.1.)
(14) If S is a faithfully flat R-algebra, then IS R = I. (Proposi-
∩
tion 1.6.2.)
(15) The integral closure of a Zn Nm-graded ideal is Zn Nm-
× ×
graded. (Corollary 5.2.3.)
x
(16) If R is a polynomial ring over a field and I is a monomial
ideal, then I is also a monomial ideal. The monomials in I are
exactly those for which the exponent vectors lie in the Newton
polyhedron of I. (Proposition 1.4.6, more general version in
Theorem 18.4.2.)
(17) Let R be the ring of convergent power series in d variables
X ,...,X over C, or a formal power series ring in X ,...,X
1 d 1 d
over field of characteristic zero. If f R with f(0) = 0, then
∈
∂f ∂f
f X ,...,X .
1 d
∈ ∂X ∂X
1 d
(cid:18) (cid:19)
(Corollary 7.1.4 and Theorem 7.1.5.)
(18) Let (R,m) be a Noetherian local ring which is not regular and
I an ideal of finite projective dimension. Then m(I : m) = mI
and I : m is integral over I. (Proposition 1.6.5.)
(19) x I if and only if there exists c R not in any minimal
∈ ∈
prime ideal such that for all sufficiently large n, cxn In.
∈
(Corollary 6.8.12.)
(20) If R is a Noetherian local ring, then ht(I) ℓ(I) dimR.
≤ ≤
(Corollary 8.3.9. The notation ℓ() stands for analytic spread.)
(21) If R is local with infinite residue field, then I has a minimal
reduction, and every minimal reduction of I is generated by
ℓ(I) elements. (Proposition 8.3.7.)
(22) If I or J is not nilpotent, then ℓ(IJ) < ℓ(I)+ℓ(J). (Proposi-
tion 8.4.4.)
(23) x In if and only if
∈
ord (xi)
I
lim n.
i i ≥
→∞
(Corollary 6.9.1 and Lemma 6.9.2.)
(24) If R S is a normal ring homomorphism of Noetherian rings,
→
then for any ideal I in R, IS = IS. (Corollary 19.5.2.)